2006-1859: A COMPARISON CASE STUDY FOR DYNAMICS ANALYSISMETHODS IN APPLIED MULTIBODY DYNAMICS

Shanzhong (Shawn) Duan, South Dakota State UniversityShanzhong (Shawn) Duan received his Ph.D. from Rensselaer Polytechnic Institute in 1999. Hehas been working as a software engineer at Autodesk for five years before he became an assistantprofessor at South Dakota State University in 2004. His current research interests include virtualprototyping, mechanical design and CAD/CAE/CAM.

© American Society for Engineering Education, 2006

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A Comparison Case Study for Dynamics Analysis

Methods in Applied Multibody Dynamics

Abstract

This paper discusses how a simple comparison case study has been utilized in an applied

multibody dynamics (AMD) course to enhance students’ learning of dynamic analysis methods

to set up equations of motion for multibody systems. The comparison case used is a planar rigid

body double pendulum with a pin joint connection between two bodies. This simple case has

helped students directly understand and see advantages and disadvantages of each dynamic

analysis method used to set up equations of motion. Based on what they have learned from this

case study, students have a better understanding of targeted dynamic analysis methods and can

more efficiently choose a proper method to analyze the motion behaviors of their design

applications than they could previously.

Introduction

An applied multibody dynamics course is usually offered to mechanical engineering

undergraduates in their senior year and to graduates in their first year. It is an advanced topic and

requires that students have a background in linear algebra, vector-matrix operations, dynamics,

numerical analysis, and fundamentals of computer science, as well as in basic programming

skills. The specific contents of multibody dynamics may vary from school to school. But

generally speaking, they may contain but are not limited to the following: (1) Multibody

kinematics: coordinate transformation matrixes and direction cosines, kinematical formulas,

partial velocities, partial angular velocities, Euler angles, Euler parameters and kinematical

differential equations, and so on; (2) Inertia: rotation of coordinate axes for inertia matrices and

principal moments of inertia; (3) Multibody kinetics: various dynamic analysis methods for

equations of motion. (4) Numerical issues in applied multibody dynamics6, 11, 12

.

In practice, many dynamics analysis methods are available for formulation of equations of

motion of a multibody system. Newton-Euler equations, Lagrange’s equations, principles of

virtual work, Hamilton’s principle, Gauss’s principle, Jordan’s principle, Kane’s method, and

even finite element methods have been used by researchers in various applications1. Three P

age 11.27.2

commonly-used methods are Newton-Euler equations, Lagrange’s equations, and Kane’s

method1, 5, 15

.

However, students may easily feel lost at such extremely mathematically-orientated methods

when they need to select a proper dynamic analysis method to set up the equations of motion for

their designs. Because they have difficulty in understanding methods, they eventually lose

confidence when they have to select a proper method for their applications.

To facilitate students’ understanding of these three methods, case study methodology, an

instructional approach widely used in various subject areas, has been utilized in the applied

multibody dynamics to help them learn how to select a proper method for virtual prototyping of

their design applications.

Applied Multibody Dynamics and Background of Students at SDSU

The dual-number course ME 592-03/492-03 applied multibody dynamics is a three-credit

technical elective course offered in the mechanical engineering program at South Dakota State

University (SDSU) to students majoring in mechanical engineering and other engineering

disciplines.

Generally, applied multibody dynamics can be structured and organized in numbers of ways. The

following are three common instructional approaches:

(1) Introducing functions, commands, user interfaces, and a user manual of commercial virtual prototyping software without having a minimal knowledge of its theoretical bases.

(2) Introducing multibody kinematics, multibody kinetics, and dynamic analysis methods for equations of motion and constraint equations but without proper use of commercial virtual

prototyping computer software.

(3) Introducing both multibody dynamics theory and computer software functions in an integrated way.

Each way has its strengths and weaknesses. The following table shows a brief comparison:

Table 1: A Brief Comparison of Three Different Ways to Organize AMD Emphasis on course

contents

Level of course Time

constraint

% of use of

software

Difficulty of

course

Software-

orientated

Workshop to train

software user

High High Low

Theory-

orientated

Ph.D. level graduate

course

Low Low High

Theory/ software

combined

College level course

for undergraduates &

1st year graduates

Middle Middle Middle

During fall 2005, undergraduates and graduates enrolling in ME492-03/592-03 came from one of

the following two groups:

(1) They had taken EM 215 dynamics, MATH 471 numerical analysis, and CSC 150 computer science I, but had not taken any advanced dynamics course yet. So they had no

background in advanced dynamical analysis methods such as Lagrangian equations.

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(2) They had taken an advanced dynamics course and at least knew Lagrangian equations.

All students in these two groups had little or no background in applied multibody dynamics and

no experience with virtual prototyping software.

Based on the technical background of the students, the approach of combining theory with the

use of software was utilized to deliver the AMD course. Such an approach has several benefits.

One obvious benefit is that students are usually attracted by the use of simulation tools. After the

instructional approach was determined, other teaching materials were chosen as follows:

(1) Textbook and reference books a) Thomas R. Kane/David A. Levinson, Dynamics Online: Theory and Implementation with

Autolev, Online Dynamics, Inc., 2000

b) Ahmed A. Shabana, Computational Dynamics, 2nd edition, Wiley, 2001 c) Jerry H. Ginsberg, Advanced Engineering Dynamics, 2nd edition, Cambridge University

Press, 1998

(2) Computer software: Autolev and Matlab. (3) Course length: Forty lectures were delivered during fall semester of 2005: three fifty-minute

lectures each week.

However, how can students be motivated to learn theory? More specifically, how can students be

motivated to proactively learn and understand various dynamic analysis methods to set up

equations of motion for their applications? In order to encourage students’ learning theory, the

AMD class exploited case study methodology in teaching three commonly-used dynamic

analysis methods: the Newton-Euler approach, the Lagrange approach and Kane’s method.

Case Study Methodology for Teaching and Learning

Case study methodology has been widely exploited as an instructional approach in various

subject areas such as medicine, law, business, education, engineering, technology, and science.

Use of this teaching method has been extensively discussed in the literature8, 9, 10, 16

.

The case study method promotes team-based activities, active learning and the ability to handle

open-ended problems10

. Case study methodology also fosters the development of higher-level

cognitive skills8, 9

. Shapiro13

summarizes several teaching and learning approaches as follow:

lectures and readings facilitate “acquiring knowledge and becoming informed about techniques”;

exercises and problem sets provide “the initial tools for exploring the applications and limitation

of techniques”; case methodology promotes the “development of philosophies, approaches and

skills”.

Case study methodology has been widely used in teaching and learning of engineering subjects.

Advantages of case study methods have been presented by Sankar et al14

in “Importance of

Ethical and Business Issues in Making Engineering Design Decision.” They concluded that the

use of the case study methodology to deal with real-world examples is highly motivating and

increases understanding of the importance of ethical issues in making engineering design

decisions. Page 11.27.4

Jensen discussed the merits of case study methodology for teaching freshman engineering

courses4. The range of engineering disciplines and contents covered were engineering analysis,

design methods, engineering calculations, technical communications and ethics. The approach

has improved students’ involvement, motivation, and interest. The outcomes of the study are

positive and promising.

Beheler et al3 specifically applied a case study approach to teaching engineering technology.

Their experiences showed that it is a viable teaching method to enhance educational outcomes

and provide students with a more meaningful and relevant academic experience. Graduating

students develop and obtain the skills and knowledge that corporate employers have reported to

be essentials to improving job seekers’ employability. Also, their experience indicated that the

approach provides a valid way to enhance problem-solving, critical-thinking, communication,

and documentation skills.

General merits of the case study approach in Barrott2 are summarized as follows:

a) Providing students with a link to the real world b) Developing students’ critical-thinking and problem-solving skills c) Developing students’ communication skills d) Involving students in a cooperative learning activity

Application of the Case Study Method to Teaching and Learning Dynamic Analysis

Methods in AMD

1. Selection of the case

A planar rigid body double pendulum connected by a pin joint was selected as the case. The

pendulum as shown in Figure 1 has joint axes at points O and P parallel to the unit vector 3n̂ .

Bodies A and B are slender uniform rods with mass Am and Bm , and length AL and BL

respectively. A torsional spring with the spring constant AK acts between body A and the ground.

1q and 2q are generalized coordinates. The basis vectors iâ , ib̂ , and in̂ )3,2,1( =i are fixed on

body A, body B and the ground respectively. A force QF is applied to point Q in the direction 1b̂ .

Figure 1: A Simple Case Study – the Rigid Body Double Pendulum

Though the double pendulum case is simple, it contains basic features that are necessary to

discuss the principles of the targeted dynamic analysis methods. For example, its generalized

Page 11.27.5

coordinate can be linear or angular, and absolute or relative. It contains two-level coordinate

transformations. The formulas for velocity and acceleration of two points fixed on a rigid body

and of one point moving on a rigid body can be applied to the same case respectively so that

students can do a cross check for their derivation and simulation. The low complexity of the case

also permits it fit into our forty-class schedule. Its simplicity brings benefits to teaching and

learning of the targeted dynamic analysis methods. In short, this case makes comparison of three

targeted dynamic analysis methods clear with less effort.

The double pendulum as a case study has been utilized for teaching and research in various

subject areas. Newberry17

used a double pendulum for students to learn and understand

Hamilton’s principle. Gulley found that a double pendulum was a useful case in learning the S-

function of Matlab18

. Swisher et al19

mentioned to use a double pendulum as a case study in an

integrated vibrations and system simulation course. Romano20

applied a double pendulum to

researching a modular modeling methodology in real-time multi-body vehicle dynamics.

2. Use of the case in ME 592-30/492-03 AMD

In the fall of 2005, the double pendulum case was repeatedly used in teaching and learning

AMD. The case and its variation were integrated with various teaching and learning scenarios.

The first use of the case was in a student homework assignment. Students were asked to derive

equations of motion out of the case shown in Figure 1 according to the Newton-Euler equations.

The purpose was to help students review what they had learned from the basic rigid-body

dynamics course. At the same time the case helped them apply new vector-matrix notations and

coordinate transformation matrix techniques in advanced dynamics to what they had learned.

The second use of the case was in class lectures when Lagrange equations were introduced. The

equations of motion in form of energy for the case were derived by the instructor to show an

analytical way to obtain them. The derivation was compared with the students’ derivation for

the same case in their homework using the Newton-Euler method.

Then the original case shown in Figure 1 was reduced into a double pendulum of two particles

connected by massless rigid links A and B as shown in Figure 2. The lectures about this reduced

case illustrated Kane’s method in detail. The key to Kane’s method lies in use of the concepts

Figure 2: A Variation of the Case Study – a Double Pendulum of Two Particles

1̂a

2â

1b̂ 2b̂

1̂n

2n̂3n̂

3b̂

3â

Qm

Pm

1q

2q

A

B

Q

P

Og

Page 11.27.6

of generalized speeds, partial angular velocities and partial velocities. These concepts were

discussed in detail in terms of the reduced case. Solution and simulation procedures using

Kane’ equations and the Autolev package were summarized for students as shown in the flow

charts of Figure 3. The simulation results produced by Autolev and Matlab for the given

geometric data and initial conditions for the reduced case are presented in Figure 4.

Figure 3: Solution Procedure Using Kane’s Equation and Autolev

-20

-15

-10

-5

0

5

10

15

20

0 2 4 6 8 10 12 14 16

Q1

(d

egre

e/se

c)

T (sec)

Generalized Coordinate Q1 vs. Time T

-10

-8

-6

-4

-2

0

2

4

6

8

10

0 2 4 6 8 10 12 14 16

Q2

(d

egre

e/se

c)

T (sec)

Generalized Coordinate Q2 vs. Time T

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16

U1

(d

egre

e/se

c)

T (sec)

Generalized Speed U1 vs. Time T

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 2 4 6 8 10 12 14 16

U2 (

deg

ree/

sec)

T (sec)

Generalized Speed U2 vs. Time T

Figure 4: Simulation Results of the Reduced Case Produced by Autolev & Kane’s Method

Select generalized

coordinates &

generalized speed

Kinematical analysis for

velocities, angular velocities,

accelerations, angular

accelerations, partial velocities

and partial angular velocities

Kinetic analysis for

generalized active

forces & generalized

inertial forces 1

1 Equations

of motion

Autolev produces

motion simulation

codes in Matlab or C

based on given

geometric data &

initial conditions

Compile & execute

Matlab code or C

code to produce

simulation data

Interpret

simulation

results

Page 11.27.7

Finally, the original case was assigned to students to derive the equations of motion by Kane’s

method and produce simulation data using the Autolev/Matlab software packages. Through four

repeated cycles of teaching and learning of the selected case, the students’ understanding and

learning of three targeted dynamic analysis methods were enhanced. At the end of four cycles,

the students and faculty generalized and summarized about the use of each method in

corresponding research areas. Further reading of journal papers and technical articles was

suggested. The appendix, which has been scanned from the AMD class materials, shows a brief

side by side comparison between Lagrange’s method and Kane’s method as used in this case.

3. Summary of the targeted methods after the repeated case study cycles

In multibody dynamics, most dynamical formulations fall into either State Space Form (SSF) or

DeScriptor (DSF) Form as shown in equations (1) and (2) respectively:

↓→↑

=

=

),,(),(IIIIII

III

qqtRHSqqtM

qq

�

�

)1(

)1(

b

a

°↓

°→

↑

=Φ

=−−

=−

0),(

0),,(),(),(

0

I

IIII

T

III

III

qt

qqtRHSqtAqqtM

qq

λ�

�

)2(

)2(

)2(

c

b

a

In equations (1) and (2), I

q and II

q are position and velocity state variables. Matrix M is the

system mass matrix, and matrix RSH is the right hand side of the equation of motion that

contains all of external loads, body loads and inertia forces associated with centripetal and

Coriolis accelerations. Due to constraints in the equation (2c), the Jacobi constraint matrix A

and Lagrange multiplier λ appear in the equation (2b).

Of the three targeted dynamic analysis methods, generally the Newton-Euler method treats each

body separately, which results in a large but sparse system mass matrix and a simple formulation.

But if not used wisely, this method may result in order n to the fourth power, O(n4),

computational complexity with respect to n number of degree-freedom of a multibody system. In

the Newton-Euler method, much effort is required to eliminate workless constraint forces.

Lagrange’s equation can automatically eliminate workless constraint forces. But this benefit can

be offset by complicated derivatives of Lagrangians, which often results in a phenomenon of

intermediate ‘swell’ and complex formulation. Generally speaking, Lagrange’s method is an

O(n3) method. Kane’s method can avoid these disadvantages and keep the advantages of both

Newton-Euler and Lagrange. It has the first order form of equation and an O(n3) computation

complexity. A comparison of labor involved in deriving the equations of motion via different

methods may be found in the reference7. The following table shows a brief comparison.

Table 2: A Brief Comparison of Three Dynamics Analysis Methods

Methods Workless

constraints

Computational

complexity

Generalized

coordinates

Complexity of

formulation

Newton-Euler Yes O(n4) No low

Lagrange Eliminated O(n3) Yes high

Kane Eliminated O(n3) Yes low

Page 11.27.8

Assessment of the Case Study Method for Teaching and Learning AMD

Homework problems, computing assignments, quizzes and exams were used to assess students’

learning and the effectiveness of the teaching of dynamic analysis methods through the case

study. In addition, team-based course projects were used to evaluate teaching and learning. Each

project team was formed by three students. The project topic was a component or subsystem of

senior design project, Mini-Baja project, or a real dynamic system that all team members were

interested in modeling, designing, analyzing and simulating. Then they would further apply what

they had learned from this case study to select a proper analysis method for their applications,

derive kinematical and force equations, set up equations of motion, and eventually produce

simulation results. Figure 5 shows the selected examples of team projects.

Figure 5: Selected Team Project Titles in AMD

Evaluation of teaching and learning was conducted anonymously. Twelve graduate students and

eight senior students took part in the survey. Table 3 shows the percentage of students who

strongly agree or agree with questions listed in the survey about course outcomes.

Table 3: Course Outcomes from the Student Survey for AMD My Learning

increased in

this course

I made progress

towards achieving

course objectives

My interest

in subject

increased

Course helps me to

think independently

about subject

I involved in

what I am

learning

% of

students

76% 75% 75% 73% 77%

Since in fall 2005 ME 492-03/592-03 applied multibody dynamics was offered for the first time

in the mechanical program, no baseline data were available to conduct a direct comparison

between teaching AMD with the selected double pendulum case and without the case. As a

relative comparison, Table 4 shows the same survey questions answered by the students who

took EM 215 dynamics, in which the rigid body double pendulum was not used as a case study at

all.

Page 11.27.9

Table 4: Course Outcomes from the Student Survey for EM 215 Dynamics My Learning

increased in

this course

I made progress

towards achieving

course objectives

My interest

in subject

increased

Course helps me to

think independently

about subject

I involved in

what I am

learning

% of

students

71% 69% 66% 69% 64%

Concluding remarks

A rigid body double pendulum and its variation have been used repeatedly as a case study for

teaching and learning through various phases of the AMD course. Though the case is simple, the

integration of the case with various educational activities provides many benefits for teaching

and learning about three targeted dynamic analysis methods used for virtual prototyping of

mechanical systems. Student surveys have provided first-hand information for further

improvement and future investigations.

Bibliographies

1. Anderson, K. S. (1990). Recursive Derivation of Explicit Equation of Motion for Efficient Dynamic/Control

Simulation of Large Multibody Systems. Ph.D. Dissertation Stanford University. UMI, No. 9108778

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Proceedings of the 2001 American Society for Engineering Education Annual Conference & Exposition.

Albuquerque, NM.

3. Beheler, A. and Jones, W. A. (2004). Using Case Studies to Teach Engineering Technology. Proceedings of the

2004 American Society for Engineering Education Annual Conference & Exposition. Salt Lake City, UT.

4. Jensen, J. N. (2003). A Case Study Approach to Freshman Engineering Courses. Proceedings of the 2003

American Society for Engineering Education Annual Conference & Exposition. Nashville, TN.

5. Hollerbach, J. M. (1980). A Recursive Lagrangian Formulation of Manipulator Dynamics and a Comparative

Study of Dynamics Formulation Complexity. IEEE Trans. Systems, Man, and Cybernetics. Vol. SMC – 10, No. 11,

November. pp. 730 – 736.

6. Huston, R. L. (1990). Multibody Dynamics. Butterworth-Heinemann.

7. Kane, T. R. and Levinson, D. A. (1980). Formulation of Equations of motion for Complex Spacecraft. Journal of

Guidance and Control, Vol. 3, No. 11. pp. 99-112.

8. Kolodner, J. (1993). Case-Based Reasoning. Morgan Kaufman, San Manteo, CA.

9. Leake, D. (1996). Case-Based Reasoning.: Experiences, Lessons, and Future Directions. AAAI Press/MIT Press,

Cambridge, MA.

10. Meyers, C. and Jones, T. B. (1993). Promoting Active Learning: Strategies for the College Classroom. New

York, Wiley.

Page 11.27.10

11. Roberson, R. E. and Schwertassek, R. (1988). Dynamics of multibody systems. New York: Springer-Verlag.

12. Shabana, A. A. (1998). Dynamics of Multibody Systems. Cambridge. Cambridge University.

13. Shapiro, B. P. (1984). Introduction to Cases. Harvard Business Online, Boston, MA. 9-584-097.

14. Sankar, C. S. and Raju, P. K. (2001). Importance of Ethical and Business Issues in Making Engineering Deisgn

Decisions: Teaching through Case Studies. Proceedings of the 2001 American Society for Engineering Education

Annual Conference & Exposition. Albuquerque, NM.

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16. Wright, S. (1996). Case-based instruction: Linking theory to practice. Physical Educator. Vol. 53, Issue 4.

17. Newberry, C. F. (2005). A Missile System Design Engineering Model Graduate Curriculum. Proceedings of the

2005 American Society for Engineering Education Annual Conference & Exposition. Portland, OR.

18. Gulley, N. (1993). PNDANTM2 S-function for Animating the motion of a double pendulum. The Math Works,

Inc.

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Albuquerque, NM.

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Technical Paper Series No. 2003-01-1286.

Page 11.27.11

Appendix: A Brief Derivation Comparison between Kane’s Equation and Lagrange’s

Equations for the Selected Case in Figure 1 (Scanned from the ME 492/592

AMD Course Materials)

From Figure 1, coordinate transformation matrixes between body A, body B & the ground are as

follows:

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